## Tower Of Twos

Let \(T(n)\) be the tower of powers of \(2\) of height \(n\). For example:

\(T(3) = 2^{2^2} = 2^4 = 16\)

\(T(4) = 2^{2^{2^2}} = 2^{16} = 65536\)

So the last digit of \(T(4)\) is \(6\).

Part A | Determine the last two digits of \(T(100)\). | |
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Part B | Determine the remainder when \(T(2021)\) is divided by \(2021\). |

## Van Erk's Sequence

Van Erk's Sequence begins in the following manner, with \(V(0)=V(1)=0\) and \(V(2)=1\).

\(V(0),V(1),V(2),V(3), \ldots = 0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, \ldots\)

Suppose the \(n^{th}\) term is \(V(n) = x\). Then the \((n+1)^{th}\) term is determined by:

- If \(x\) has not appeared in the sequence before \(V(n)\), then \(V(n+1) = 0\).
- Otherwise, if the \(k^{th}\) term is \(V(k) = x\) and this is the last appearance of \(x\) in the sequence before \(V(n)\), then \(V(n+1) = n-k\).

Part A | Determine \(V(123)\) | |
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Part B | Determine \(V(1234567)\) |